From Kyle to Glosten & Milgrom and Back Umut Çetin (based on a joint work with Hao Xing) London School of Economics Paris, 18 June 2014 U. Çetin From Kyle to G-M and Back
Kyle’s model of insider trading Back (1992) formalised the continuous time version of the model first introduced by Kyle (1985): 1 Noise traders: The noise traders have no information about the future value of the risky asset and can only observe their own cumulative demands. The cumulative demand is modelled by a standard Brownian motion denoted with B . 2 Informed trader: The insider “knows” the value of the risky asset, f ( V ) , at time 1, where V is a standard normal random variable independent of B . Being risk-neutral, her objective is to maximize her expected profit. 3 Market makers: The market makers observe the total order and compete in a Bertrand fashion to clear the market. They are assumed to be risk-neutral. U. Çetin From Kyle to G-M and Back
The pricing mechanism of the market The market makers decide the price looking at the total order, Y , which is given by Y t = B t + θ t , where θ t is the position of the insider in the risky asset at time t . Thus, the common filtration of the market makers is the one generated by Y and is denoted with F Y . Note that θ is not necessarily adapted to F Y , i.e. the insider’s trade is not observed directly by the market makers. A market maker has a pricing rule , H : [ 0 , 1 ] × R �→ R , to assign the price in the following form: P t = H ( t , Y t ) , where P t denotes the price quoted by the market maker for the net demand at time t . U. Çetin From Kyle to G-M and Back
Objectives of the insider and the market maker The pricing rule is assumed to be strictly increasing in demand leading to the conclusion that the insider has the full information. U. Çetin From Kyle to G-M and Back
Objectives of the insider and the market maker The pricing rule is assumed to be strictly increasing in demand leading to the conclusion that the insider has the full information. The market maker chooses a rational pricing rule, i.e. he chooses a pricing rule so that H ( t , Y t ) = E [ f ( V ) |F Y t ] , for every t ∈ [ 0 , 1 ] . The insider aims to maximize her expected profit out of trading. If X is continuous and of FV, the profit at time 1 equals � 1 � 1 X t dP t + X 1 ( f ( V ) − P 1 ) = { f ( V ) − P t } dX t . 0 0 U. Çetin From Kyle to G-M and Back
Avenues for a robust approach if one must! V is the private signal of the informed trader. There is no a priori reason that everyone knows the mean and the variance of this normal random variable. Also the noise traders trade for reasons exogenous to the model. Thus, their volatility is not necessarily known. Both extensions will require some theory of robust filtering (possibly beyond what is already known to the statisticians), which is interesting in its own. In this talk I will ignore the impacts of above kinds of ambiguities. U. Çetin From Kyle to G-M and Back
Equilibrium Definition 1 A pair ( H ∗ , θ ∗ ) is said to form an equilibrium if H ∗ is a pricing rule, θ ∗ ∈ A , and the following conditions are satisfied: 1 Market efficiency condition: Given θ ∗ , H ∗ is a rational pricing rule. 2 The optimality condition: Given H ∗ , θ ∗ maximizes the expected profit of the insider. U. Çetin From Kyle to G-M and Back
Optimality conditions for the insider’s problem Working out the formal HJB computations yields that in an equilibrium H should satisfy H t + 1 2 H yy = 0. Following Back the trading strategies that are not absolutely continuous are necessarily suboptimal. U. Çetin From Kyle to G-M and Back
Optimality conditions for the insider’s problem Working out the formal HJB computations yields that in an equilibrium H should satisfy H t + 1 2 H yy = 0. Following Back the trading strategies that are not absolutely continuous are necessarily suboptimal. Therefore, it is enough to consider X ∈ A of the form dX t = α t dt for some measurable α . With an abuse of notation, the insider’s problem becomes: �� 1 � E v sup { V − H ( t , Y t ) } α t dt , (1) 0 α ∈A U. Çetin From Kyle to G-M and Back
Optimality conditions for the insider’s problem (cont’d) The value function can be computed explicitly: � y � 1 { H ( t , u ) − V } du + 1 J ( t , y ) := H y ( s , ξ ( s )) ds 2 ξ ( t ) t where ξ ( t ) is the unique solution of H ( t , ξ ( t )) = V . Direct calculations lead to J t + 1 2 J yy = 0 so that �� 1 � = E v [ J ( 0 , 0 )] − X ∈A ( H ) E v [ J ( 1 , Y 1 )] . E v sup ( V − H ( t , Y t )) dX t inf 0 X ∈A ( H ) Thus, the optimal strategy for the insider is to find an absolutely continuous X so that H ( 1 , B 1 + X 1 ) = V . U. Çetin From Kyle to G-M and Back
Equilibrium The equilibrium level of the total demand, Y ∗ , satisfies t = dB t + V − Y ∗ dY ∗ t 1 − t dt , so that it is a Brownian bridge. The price is given by P t = H ( t , Y ∗ t ) , where Y solves H t + 1 2 H yy = 0 , H ( 1 , y ) = f ( y ) . Y ∗ is a Brownian motion in its own filtration. Related literature: Wu (1999), Cho (2003), Laserre (2004), Biagini and Øksendal (2005). U. Çetin From Kyle to G-M and Back
An arbitrage strategy In the original Kyle model f ( V ) = V so the profit of the insider at time 1 equals: � 1 ( V − Y t ) 2 dt > 0 . 1 − t 0 The above is an arbitrage but due to feedback effects, it cannot be scaled, hence an equilibrium, in which the insider has a finite wealth, exists. In the original Kyle model the insider buys when the asset is undervalued, i.e. V − Y t ≥ 0, and sells otherwise leading to an ‘arbitrage’ in every infinitesimal trade. U. Çetin From Kyle to G-M and Back
This is not always the case for any f since in general the final wealth is given by � 1 ( V − Y t )( f ( V ) − H ( t , Y t )) dt , 1 − t 0 and V − y ≥ 0 is not equivalent to f ( V ) ≥ H ( t , y ) . The above considerations show that the insider does not trade based on whether the market price is higher than her own valuation but rather on whether her own signal, V , is more optimistic, i.e. ≥ , than the signal of the market makers, Y . U. Çetin From Kyle to G-M and Back
Glosten-Milgrom Model Traders submit a fixed size of δ buy or sell orders for the asset and f ( V ) takes values from the set { 0 , 1 } . The total demand of the noise traders, Z , is given by the difference of two pure jump processes N B and N S . N B /δ and N S /δ are assumed to be independent Poisson processes with constant intensity β . Thus, the noise trades still follow a martingale. The net order of the insider is denoted by θ := θ B − θ S where θ B (resp. θ S ) denotes the cumulative buy (resp. sell) orders. The market makers choose a Markovian pricing rule to set the price as before. U. Çetin From Kyle to G-M and Back
Unlike the Kyle equilibrium, in this model the insider uses a mixed strategy. The characterisation of equilibrium is similar to that of Kyle’s in the sense that the insider trades without changing the distribution of the total demand while making sure that the market price matches to the announcement value at time 1. U. Çetin From Kyle to G-M and Back
Unlike the Kyle equilibrium, in this model the insider uses a mixed strategy. The characterisation of equilibrium is similar to that of Kyle’s in the sense that the insider trades without changing the distribution of the total demand while making sure that the market price matches to the announcement value at time 1. In this special case this amounts to the insider adding a ‘drift’ to the noise demand which will vanish when projected on the filtration of the market makers but at the same time ensure that [ Y 1 ≥ y δ ] = [ f ( V ) = 1 ] where y δ is such that P ( Z 1 ≥ y δ ) = P ( f ( V ) = 1 ) . In Kyle’s framework the way to guess the optimal strategy is to apply the initial enlargement of filtration formula to Brownian motion with its time-1 value and the so-called ‘information drift’ will be the optimal choice for the insider. U. Çetin From Kyle to G-M and Back
If we apply the same recipe here, taking δ = 1 we obtain that the intensity of Z B , when the natural filtration of Z is enlarged with the set [ Z 1 ≥ y 1 ] , becomes 1 [ Z 1 ≥ y 1 ] β h ( Z t − + 1 , t ) + 1 [ Z 1 < y 1 ] β 1 − h ( Z t − + 1 , t ) , (2) h ( Z t − , t ) 1 − h ( Z t − , t ) where h ( z , t ) := P [ Z 1 ≥ y 1 | Z t = z ] . A similar formula for the intensity of Z S yields that the information drift is again absolutely continuous. This is problematic since a continuous trading will reveal the identity of the insider at once and the market makers will immediately adjust the market price to the true price causing the insider loosing her informational advantage. U. Çetin From Kyle to G-M and Back
The optimal strategy for the insider The optimal strategy for the insider is to find two point process θ B and θ S so that the intensity, in her own filtration, of Z B + θ B (resp. Z S + θ S ) equals to that of Z B (resp. Z S ) when the natural filtration of Z is enlarged with the set [ Z 1 ≥ y 1 ] . This is how the mixed strategy arises. U. Çetin From Kyle to G-M and Back
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